Lumped-Parameter Model of the Delay Solenoid Valve

Transcription

1 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. IE-29, NO. 3, AUGUST 1982 Lumped-Parameter Model of the Delay Solenoid Valve with Integral Thermistor WILLIAM G. HURLEY, MEMBER, IEEE 225 Abstract-In the past, many applications have been found for delay solenoid valves using an integral thermistor for delay action. These applications include oil and gas flow control in hydraulic and pneumatic systems used in a wide variety of situations, ranging from the residential heating furnace to large industrial controllers. Because of the large number of parameters involved in the design of these valves, no known analytical method exists to accurately predict time delay and power consumption. This has resulted in a trial-and-error approach to design and thermistor selection involving expensive prototypes for testing purposes. In this paper the author has developed an accurate lumped-parameter model. This model clearly illustrates those parameters which govern the terminal variables such as power consumption and also allows thermistor selection for a given delay based on manufacturers' specifications. The scope of applications of delay valves should be increased as a result of greater predictability in valve performance and greater flexibility and reduced cost in thermistor selection. The principle of geometric similarity is also established, which leads to scaled-down prototypes of larger valves for testing purposes. I. INTRODUCTION THE BASIC electromechanical system employed in a typical delay solenoid valve is shown in Fig. 1. It consists of a thermistor in series with a coil, wound on a ferromagnetic tube. The tube constrains the movement of the plunger. Current flowing in the coil sets up a mechanical force of electric origin which attracts the plunger into the tube to allow fluid flow. The magnetic force on the plunger is opposed by a spring which closes the valve when the system is de-energized. A certain value of current is required to counteract the spring force and open the valve. As soon as the system is energized, a current flows which is insufficient to open the valve. This causes the negative temperature coefficient thermistor to heat up which in tum reduces its resistance, allowing more current to flow. Eventually, the current will increase to a value which opens the valve and allows fluid to flow through it. This situation is complicated by the fact that heat is also generated in the coil resistance and in the ferromagnetic tube and plunger. In order to set up a consistent model, the electromechanical system must be modeled to predict the current transient and the thermal system must be modeled to predict thermistor resistance as a function of time. The two models are combined to predict time delay. The steady-state power consumption can also be predicted using these models. The key to the modeling process is the heat balance between the thermistor and its surroundings. The thermistor temperature controls its resistance which in turn controls the current in the system that sets up the electric force which Manuscript received August 5, 1980; revised November 30, The author is with Ontario Hydro, Toronto, Ont. M5G 1X6, Canada. Fig. 1. Electromechanical system of the delay valve. opens or closes the valve. Because of the complicated relationships involved, a finite-difference method is used to solve the governing differential equations. II. LUMPED-PARAMETER MODEL DEVELOPMENT Two equivalent circuits of the delay valve are shown in Fig. 2: Fig. 2(a) shows a parallel combination to represent the core, while Fig. 2(b) shows the equivalent series combination. Rt is the thermistor resistance which is a function of temperature. R, is the coil resistance and R,0 represents the core losses equivalent series resistance. L, is the coil self-inductance which is a function of plunger position. The internal coil voltage Eo will depend somewhat on the variations in R, and Rt during operation, but these changes are insignificant. Magnetic nonlinearities are neglected since the terminal voltage is constant in any given application. In the following sections, each parameter will be examined and relationships established. Coil Self-Inductance Lc By considering the magnetic field system of Fig. 1, it can be easily shown that the inductance varies inversely with plunger position [11 x 1 - g /82/ $ IEEE Lo= APo2 g where Lo is the value of L.(x) for x = 0, i.e., with the valve open (Fig. 1). g is a constant which depends on the gap between the plunger and tube; x is the plunger position;, is the magnetic permeability of the plunger material; Ap is the plunger cross-sectional area; and N is the number of coil turns.

2 226 +I vo R L gco bm -( (a) IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. IE-29, NO. 3, AUGUST 1982 where M is the mass and Cp the specific heat of the thermistor. 60 is used to indicate free air cooling. Integrating the above equation for thermistor initial temperature T1 gives so T- Ta = (T1 -T)e-"T, r060 = MCp MCp To = o0 IC Eo Rt+ RC)O Imovo~~~~~~v Im 0 (c) Fig. 2. Equivalent circuits and phasor diagram of the delay value. (a) Parallel circuit. (b) Series circuit. (c) Phasor diagram. In practice, Lo and g are found from measurements of coil self-inductance in the valve open and closed positions. Equivalent Core Resistance R,o The core losses consist of hysteresis and eddy current losses, and so Rco would be a function of plunger position with a relationship similar to that for L(x). Before the valve opens, the thermistor resistance is very large compared to Rco. We require the value of Rco for the closed valve to determine delay, we require the open-valve value to determine steadystate core losses. These two values are sufficient for modeling purposes. Thermistor Resistance Rt For negative temperature coefficient thermistors, the resistance R is related to the absolute temperature T by [2] R R-oefl[(I1 T)-( 1 TO) l where Ro is the resistance at temperature To(K), and,(k) is a constant of the thermistor. Normally, Ro is specified at 250C. Ro and j3 are given in the manufacturer's specifications. For a voltage E and current I applied to the thermistor, the dissipation constant is [3] 6 = EI T- Ta MX/OC. EI is the thennistor internal heat generation, and T is the thermistor temperature with ambient temperature Ta. Besides being a function of ambient temperature, the dissipation constant also depends on the mounting method. The heat balance for a thermistor cooling in air is [31 -MCpdT = 6 o(t -Ta) dt where T0 is the time constant for the above cooling process. Thus the time constant is also a function of mounting of the thermistor, but the product r6 is a constant for any given thermistor. Usually, the manufacturer supplies r0 and 60 so that for any given application, r or 6 can be measured and the other found from the above relationship since 1T6 1 = T00. Coil Resistance RC The coil resistance is a function of coil temperature given by RC =R25 [ (T- 25)] where R2 5 is the resistance at 250C and a!2 5 is the corresponding temperature coefficient of resistance. For copper a2 5 = /0C. Note that the coil has a positive temperature coefficient of resistance in opposition to the negative coefficient of the thermistor. In general, for a wire of length L and cross section a we have L R =p - a where p is the conductor resistivity. Thus the resistance is proportional to the number of coil turns. If we maintain the same volume of conductor in the coil, and increase the turns by a factor x, then the conductor cross section is reduced by x so that the coil resistance will vary with x2 or the square of the turns ratio R2 N2 2 RI N, III. STEADY-STATE ANALYSIS At any instant during valve operation for an applied voltage VO and current Io, the following relationships hold (refer to Fig. 2 for phasor diagram): Power factor Consumed power Induced voltage Core loss Core loss component of current = cos o Po = VolO cos 0 Eo = Vo-Io(Rc +Rt) Pco =Po-(Rc+R yo)2 ICQ= I CO =Pco

3 HURLEY: LUMPED-PARAMETER MODEL OF DELAY SOLENOID VALVE 227 Core conductance Magnetizing component of current Magnetizing susceptance Core resistance Core reactance Pco g0o =E2,MO -1o Rb Imo o = Eo 'co bmo =gco2 + bo2 IV. TRANSIENT ANALYSIS Dynamic Model-Current to Open I, p The magnetic co-energy of the system in Fig. 1 is Wm' = f X(i' x) di'. SinceX=LiandL,(x)=Lo/(I +x/g) we obtain [1I Wm'- WIn L0i2 o 2(1 + x/g) The force of electric origin is then Coil inductance L -- 2rrf So by measuring PO, VO, Io, and R, and Rt, all other parameters can be found. Rt should be measured immediately after power is removed since its value changes quickly. f is the frequency of the applied voltage (in units of hertz). After the valve opens, the thermistor continues to heat up until an equilibrium is established between the heat generated in the valve coil and thermistor and the heat dissipated to ambient. If Po, VO, Io are measured in the steady state after the valve opens, then Lo and R,o can be found since x = 0. If the valve is forced closed, and again PO, VO, Io measured, then R,o = RC0C in the valve closed position is found (the thermistor can be removed for this test to facilitate readings). Also, gis found from Lo Xc I + XC/9 where Xc is the value of x, the plunger position when the valve is closed. Finally, the steady-state losses are Coil copper losses P_ =RcJ 2 Thermistor dissipation Core loss Pt =R to2 Pco = R oio2 Total dissipation PV = PC Pt + PcO. In these equations, Rt and Rc are steady-state values. Just after the valve opens, the coil is still close to its ambient temperature. Since the thermistor has a negative temperature coefficient of resistance, and the coil has a positive coefficient, their sum tends to remain constant during the transient. Thus the steady-state total power dissipation may be estimated by measuring Rc when the coil is cold, and measuring Rt just after the valve opens. This was found to give good results in practice. For this calculation, the value of current is that which is required to open the value Ip, which will be calculated in the next section. fe = awm' ax Loi2 2g(1 +±Xlg)2 Applying Newton's Law to the plunger, neglecting damping we have for a current Io d2x M- +K(x-i)= _too L0102 dt2 2g(1 +x/g)2 fs(x) = -K(x - 1). fs(x) is the spring force acting on the plunger, K is the spring constant, and I is the value of x for which the spring force is zero. For stable equilibrium d2x M- =0 dt2 so K(x -l2)g+ ± / 0. 2g(1 + Xlg)2 The graphic solution of this cubic equation is shown in Fig. 3. At X1 and X2 the plunger is statically balanced. The third root has a negative value for x. It is obvious from the graph that X1 is dynamically unstable and that X2 is dynamically stable. Perturbations at XT to reduce x cause it to be further reduced by the electric force; on the other hand, a perturbation to increase x at X2 results in the plunger returning to X2 under the influence of the electric force. This can be shown mathematically by comparing the derivatives of the force functions given above at X1 and X2. In Fig. 3, curves A and B illustrate situations where no statically stable points exist and where one statically stable point exists, respectively. In a delay solenoid valve, when the system is first energized, the current is limited by the thermistor, and curve B describes the system. As the thennistor heats up, the current increases and the dynamically unstable equilibrium point at X1 moves towards Xc: the value of x corresponding to the valve closed

4 228 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. IE-29, NO. 3, AUGUST 1982 Fig. 3. Ope n Closed Xc Equilibrium points for the delay valve. position. When X1 = Xc, the valve opens and curve C applies. The current continues to increase, and eventually reaches a steady-state value represented by curve A and the valve remains open. When the valve opens, the current is reduced due to the increased inductance at x = 0 (Lo). As long as the electric force at x = 0 corresponding to this reduced current is greater than the spring force, the valve remains open. Solving the cubic equation for x = Xc gives the current required to open the valve Iop = K(l XC)2g(1 +Xc/g)2 1/ 2 Lo ThermalModel-Time to Open Td Having found Iop, we now wish to determine the time taken to reach that value. As stated in Section II, the dissipation constant and the time constant for the thermistor are dependent on the method of mounting. As an integral thermistor, there is some electrical insulating material between the thermistor and the coil. Since the time constant of the copper coil is much greater than that of the thermistor, we treat the coil as a heat sink at ambient temperature for transient analysis. The dissipation constant 6, for the thermistor mounted in the valve may be found as follows; Apply a voltage E (ac or dc) to the thernistor in the valve and measure the steady-state current I, the resistance R is R = E/I so the temperature T can be found from R =ROefN(lT)-(/To)J and EI T- Ta where Ta is the ambient temperature during the test. From Section II, the thermistor time constant is TV = 0oro 6 o, ro, Ro, and,b, are given by the manufacturer. From Fig. 2 and our equivalent circuit in Section III, the current in the coil just before the valve opens is where Xc is VO RC + RC0C ±Rt(t) +j2tflc(xj) the value of x in the valve closed position as before. The heat generated in the thermistor is Pt = RtIo(t)2 so at time t, the heat balance for the thermistor is Pt(t)dt =MCpdT+ v(t- Ta)dt. (1) Combining the above expressions for R(T), Pt, and IO(t), obtain Roelf(l/T)-(l/TO) V02 [RC ± Rcoc +RoeP[(1I T)- (1 ITO) I] 2 + [ 27rfLC(Xc)] 2 d ==MCpdT + 6v(T-Ta) dt. This is a very complicated differential equation for thermistor temperature as a function of time. To simplify the solution, we introduce a finite-difference type solution. We assume Pt is constant in the time interval (ti-t1) and integrate (1) to obtain ±Pt(t)1)t" ita = (T-Ta)e titu + t(1 _eat/r) 5u where At = t1 - ti. By taking sufficiently small time intervals, acceptable accuracy is obtainable. In the interval ti-tj, we have Rti = Roe fl I Ti3 -(i ITO) I I- ~~~V0 oi = RC + RcOc + Rti +!2iTf Pti = Rtijoi2 c(xc) (Tj-a= (Ti - T)e-T Jr (i(-e-lv t= ti + At. The initial conditions are T1 = Ta, t1 = 0. When Ioi becomes equal to Iop calculated in the last section, the elapsed time is equal to Td, the valve delay. The above method is amenable to solution on a programmable hand-held calculator. V. EXPERIMENTAL RESULTS AND DESIGN EXAMPLE The above model was applied to two valves which had completely different physical characteristics and also had different thermistors. The results are shown in Table I. we

5 HURLEY: LUMPED-PARAMETER MODEL OF DELAY SOLENOID VALVE TABLE I EXPERIMENTAL RESULTS Valve 1 Valve 2 Calcu- Calculated Measured lated Measured Power at open Pop (W) 8.31 _ 4.0 Steady state power Pv (W) In general, there is excellent agreement between theoretical predictions and practical measurements. It is interesting to note that the valve power when it is opened tumed out to be a very good estimate of the steady-state power as discussed in Section III. Let us redesign Valve 1 to reduce its power consumption while maintaining the same time delay. We assume that the same volume of copper is to be maintained. From our lumped parameter model in Section II Lo OcN2 Rco ocn2 Rc o N2 Thus by increasing the number of turns we can find the new values of the above parameters. The model is used to find the new value of time delay and power consumption. Increasing the number of turns with smaller wire reduces the overall power consumption but increases the time delay. Having fixed the turns for the correct dissipation, we must choose a new themistor to give the original delay. In Section IV, we found that the delay was proportional to the product r8 = MCp, the thermistor heat capacity. We stipulate that the delay is also proportional to Ro since it limits the initial current in the coil. In general, the delay is TdccROMCp or TdocRor. These values are supplied by the manufacturer. If we want a thermistor to give half the time delay then we must choose one with a product Ro0r reduced by one half. This techniqu proved very successful. This procedure eliminates the tediou and costly task of building a new sample for each thermistoi being considered. Note that the power consumption will no, change very much if the thermistor is changed since it maker the least contribution to overall power consumption comparec with the coil and core losses. VI. CONCLUSIONS A model has been developed which simulates the operatior of the delay solenoid valve. Practical measurements have shown that the model is very accurate in predicting the twc most important parameters: power consumption and time delay. The real strength of the model lies in its ability to check various design modifications without resorting to costly prototype models. Another area where the model is of immense value is in the quality and process control for the valve. From the model, the two values most likely to vary are the thermistor resistan-ce and the core equivalent resistance. As stated in Section II, the equivalent core resistance is negligible when the valve is closed so that fluctuations in time delay would probably be due to variations in the thermistor resistance. On the other hand, core losses make a significant contribution to overall power losses, therefore, loss variations are most likely due to changes in the valve magnetic material. In all, over 15 parameters are used in the model, which means that every component in the electromechanical system of the valve can be studied and fine tuned for optimum design. ACKNOWLEDGMENT Appreciation is extended to K. R. Cribb and members of the Evaluation Laboratory at Honeywell Limited, Canada. REFERENCES [11 H. H. Woodson and J. R. Melcher, Electromechanical Dynamics. New York: Wiley, chs. 1, 2, 3, and 5. [21 F. J. Hyde, Thermistors. London: Iliffe, 1971, ch. 2. [31 Electronic Industries Association, Standard RS-275-A, Thermistor Definitions and Test Methods, Washington, DC, (